ISSN 2070-0482, Mathematical Models and Computer Simulations, 2017, Vol. 9, No. 4, pp. 423–436. © Pleiades Publishing, Ltd., 2017. Original Russian Text © T.G. Elizarova, D.S. Saburin, 2017, published in Matematicheskoe Modelirovanie, 2017, Vol. 29, No. 1, pp. 45–62. Application of the Regularized Shallow Water Equations for Numerical Simulation of Seiche Level Oscillations in the Sea of Azov T. G. Elizarova a, * and D. S. Saburinb, ** aKeldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia bFaculty of Physics, Moscow State University, Moscow, Russia *e-mail: [email protected], **e-mail: [email protected] Received March 28, 2016 Abstract—A mathematical model for calculating the currents in the sea area scale was developed for the first time within an algorithm of regularized shallow water equations. The model and the numeri- cal algorithm are described as applied to the topology and natural features of the Sea of Azov. The results of the calculations of hydrodynamic currents in the Sea of Azov in the presence of typical seiche waves caused by tidal or wind influences are presented. Keywords: regularized shallow water equations, difference algorithm, seiche waves, the Sea of Azov DOI: 10.1134/S2070048217040044 INTRODUCTION The Azov region is a strategically important region for the Russian Federation: it has enormous trans- port, industrial, recreational, strategic, and military significance. Therefore, forecasting the dynamics and circulation of the sea at varied environmental impacts, primarily due to weather variations, is considered to be a priority problem. The Sea of Azov is distinguished for its unique topography and climate. Some climatic phenomena, mostly caused by strong winds, can bring about serious risks for people and develop to the scale of disasters. They include tidal and wind-generated oscillations of the sea level, storm winds caused by cyclonic activity, storm waves, seiches, tsunamis, and wind waves. Each of these phenomena imposes its requirements on the numerical simulations elaborated to study and forecast them. Seiches are standing waves emerging in an enclosed or partially enclosed body of water under the action of atmospheric pressure variations, winds, or storm surges from neighboring basins. In the shallow Sea of Azov, seiche waves are frequent. Currents emerging due to seiches set the total water mass of the basin in motion. At the nodal points with an almost constant water level and in narrow spots, seiches can induce extreme current velocities of up to 1.5 m/s. The amplitude of level oscillations can exceed 1 m. Seiches can significantly enhance wind-generated effects in this region, and induce cat- astrophic water level differentials. The detailed description of these phenomena in the Sea of Azov is pre- sented, for example, in [1]. Therefore, studying and forecasting seiche currents in the shallow Sea of Azov with its gently sloping shores are quite relevant. At present, there are some highly precise simulations describing the Azov hydrodynamics presented in [2–9] and references in them. These numerical simulations use various two-dimensional, three-dimen- sional, single-layer, and two-layer finite-difference algorithms, which are solved by various numerical methods, including the explicit and implicit finite-difference approximations, and the application of spaced and nonuniform grids and finite element methods. The approach offered by the authors is based on two-dimensional shallow water equations. A new numerical method for shallow water equations was offered and tested in [10] based on classical equation smoothing over a small time interval. This procedure leads to the creation of regularizing additives, which ensure the stability of the numerical solution of the problem for a wide range of parameters. This approach is expanded to nonstructured grids and can be naturally processed in parallel over the computational clus- ter. An important advantage of the algorithm is its generalization for the cases of flows that promote the 423 424 ELIZAROVA, SABURIN y 300000 250000 −4 −6 −9 200000 −5 −12 −8 − 150000 −4 6 −12 100000 −12 −11 −9 50000 −5 0 100000 200000 300000 x Fig. 1. Azov seabed topography (depth indicated in m). emergence and disappearance of dry dry bottom areas; i.e., they generate the so-called drying and flood- ing zones [11]. The approach was used, in particular, for the numerical simulation of liquid fluctuations in freighter reservoirs [12, 13] and simulation of the experimentally observed formation of a soliton on a water surface under the impact of wind in the annular tunnel [14]. In this paper, the regularized shallow water equations are used for the first time for the numerical sim- ulation of currents in the sea area scale. Calculations of the standard for the Sea of Azov seiche waves are presented as an example. Under natural conditions, these oscillations most frequently emerge due to the persistent pressure of a constantly directed wind, which shapes the initial gradient of the sea surface level. 1. STATEMENT OF THE PROBLEM IN THE SHALLOW WATER EQUATIONS One of the features of the sea and ocean hydrodynamic problems is that the aquatic environment layer is quite thin and its depth is much smaller than its longitudinal dimensions. This is widely used for building baroclinic circulation models in the seas and the entire world ocean (see, for example, papers [15, 16] and the bibliography to them). However, a simpler hydrodynamic approach of shallow water is suitable for describing some problems [17]. Within this approach, the vertical component of the flows velocity in the layer is neglected, and the longitudinal velocities are assumed constant over its depth. We consider a two-dimensional set of shallow water equations in flux form. We take into consideration the force of the wind, the Coriolis force, and the seabed friction force as external forces. Having in mind these forces and the topology of the seabed, we can write the following system: ∂ ∂uh ∂uh h ++=x y 0, ∂∂tx ∂ y ∂uh ∂∂22 ∂ ,, x +++(hu1 gh) () hu u =−+τ−τ hfcxwxb u gh b , (1) ∂∂txxxyy2 ∂ y ∂ x ∂uh ∂∂ ∂ y +++=−−+τ−τ()hu u( hu221 gh) hfcywyb u gh b ,, . ∂∂txxy ∂ y y2 x ∂ y MATHEMATICAL MODELS AND COMPUTER SIMULATIONS Vol. 9 No. 4 2017 APPLICATION OF THE REGULARIZED SHALLOW WATER EQUATIONS 425 z h h + b b x Fig. 2. Outline of variable shallow water equations. Here hxyt(,,) is the liquid height above the seabed level, uxytx(,,) and uxyty(,,) are the components of the flow velocity, g is the gravitational acceleration and function bxy(,) describes the seabed topography (Figs. 1, 2). Projections of the force of wind friction against the water surface are designated as τw and calculated τ=γiw, =+22 γ as WWi , where Wi is the wind component, WWWxy is the wind velocity module, and is the coefficient of wind friction against the free water surface. Projections of the force of water friction against the seabed are designated as τb and calculated by rela- τ=μib, μ =+22 tion uui , where is the friction coefficient and uuuxy is the flow velocity module. Friction coefficients are preset values, and for sea areas they are μ=2.6 × 10−3 [8] and γ=3.25 × 10−6 [16]. Wind velocity is also set based on the in-field observations and can be time dependent. The right parts of the equations of motion include the Coriolis force f cor with components π ffucor = c and ffucor =− c , where f =Ω2 sin φ= 22 sin φ is the Coriolis parameter, Ω=2 πT = xyyxc T −− 7.2921× 1051 s is the angular speed of the Earth’s rotation, T = 86400 is the diurnal period of the Earth’s rotation measured in s, and φ is the point latitude in degrees counted from the equator. The scope of the problem represented in Fig. 1 covers the Sea of Azov area, the Kerch Strait, and the adjacent part of the Black Sea. The inclusion of the Kerch Strait in the scope allows us to evaluate the impact of the seiche waves of the Sea of Azov on the surface levels and currents in the zone of the strait. The studied region is located within the limits from 34° 45' 6'' E to 39° 29' 38'' E, and from 44° 48' 4'' N to 47° 1612' '' N, respectively. The seabed topology is studied set on the grid with a resolution of 8'' . Due to the rather small linear dimensions of the Sea of Azov in relation to the Earth’s radius, the problem is set in Cartesian coordinates. A uniform rectangular grid with a resolution of 250 × 250 m is used. The coast- line corresponding to the undisturbed sea level is chosen as the zero mark. The observations show that in the Sea of Azov, the impact of a long-term (for several days) unidirec- tional wind can generate a surface level gradient, whose destruction produces a seiche. This seiche is an analog of a standing wave inside a pool. Below an example is given of the calculations of the surface level’s evolution and the current velocity for the standard seiche wave with the initial amplitude of one meter. 2. REGULARIZED SHALLOW WATER EQUATIONS The regularization method mentioned above in the Introduction applied to Navier–Stokes and Euler equations provides an opportunity to write effective numerical algorithms for their solution, which are stated, for example, in [18–20]. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS Vol. 9 No. 4 2017 426 ELIZAROVA, SABURIN The regularized shallow water equations and the technique of their formulation are described in [10]. Below we present the obtained equations taking friction forces and Coriolis force into account. Regular- ized equations (1) appear as ∂ ∂j ∂j h ++=mx my 0, ∂∂tx ∂ y ∂∂Π2 ∂∂hu j u jumy x∂∂⎛⎞gh cxwxbb ∂Π yx ,, xmxx+++⎜⎟ =hfug*,() −+++τ−τ xx (2) ∂∂tx ∂∂ yx⎝⎠2 y ∂∂∂ xxy ∂∂ ∂2 ∂Π∂Π huymxymyy j u j u ∂∂⎛⎞gh ⎛⎞cywybb xyyy,, + + +⎜⎟ =−−+++τ−τhfug*.⎜⎟ ∂∂tx ∂∂ yy⎝⎠2 ⎝⎠x ∂∂∂ yxy Parameters jmx and jmy have the physical sense of a regularized liquid flux and are expressed as =− =− jhuwjhuwmx(),(), x x my y y (3) where hui is the liquid flow within the shallow water approximation, and wi is the regularizing correction of velocity expressed as τ ⎛⎞∂()hu2 ∂()hu u ∂+()hb wgh=++⎜⎟x xy , x hx⎝⎠∂∂ y ∂ x (4) ⎛⎞∂∂()()2 ∂+ =++τ huxy u hu y ()hb wghy ⎜⎟.